Approximation and regularization of problems of the optimal control of the coefficients of parabolic equations
type: Item
instance of: scholarly article
title: Approximation and regularization of problems of the optimal control of the coefficients of parabolic equations
zbMATH Open document ID: 0803.49006
author: F. V. Lubyshev
published in: Computational Mathematics and Mathematical Physics
publication date: 1994-07-24T00:00:00Z
review text: The author considers an optimal control with cost functional \[ J(u,g)= \alpha_ 1 \int_ Q | u(x,t)- u_ 0(x,t)|^ 2 dx dt+ \alpha_ 2 \int_ \Omega | u(x,T)- \psi(x)|^ 2 dx \] and state equation with control in the coefficients \[ \begin{cases} u_ t- D_{x_ 1}(g_ 1(x,t)D_{x_ 1} u)- D_{x_ 2} (g_ 2(x,t)D_{x_ 2}u)+ g_ 3(x,t)u= f(x,t) &\text{in }Q\\ u(x,t)= 0 &\text{on }\partial\Omega\times (0,T)\\ u(x,0)= \phi(x) &\text{on }\Omega.\end{cases} \] Here, \(\Omega\) is a rectangle in \(\mathbb{R}^ 2\), \(Q= \Omega\times ]0,T]\), \(f\), \(\phi\), \(u_ 0\), \(\psi\) are given, and \(g= (g_ 1,g_ 2,g_ 3)\) is the control variable which is assumed to satisfy suitable boundedness conditions on the derivatives. An existence result for optimal pairs is then easily proved. The main goal of the paper is to study finite-dimensional approximation schemes for the optimal control problem above, as the difference method or the projection-difference method. A priori bounds are obtained for the error with respect to the state, and the rate of convergence of approximations is estimated.
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49M25
zbMATH DE Number: 584216
zbMATH Keywords: cost functional
zbMATH Keywords: state equation
zbMATH Keywords: projection-difference method
reviewed by: Giuseppe Buttazzo
MaRDI profile type: Publication
Recommended article: Approximation and regularization of optimal control problems for parabolic equations with controls in coefficients
Recommended article: Q3341793
Recommended article: On approximate solution of optimal control problems by parabolic equations
Recommended article: Q3779315
Recommended article: Q3505968
Recommended article: Difference approximation and regularization of a parabolic optimal control problem
Recommended article: Q3713028
Recommended article: Q3324524
Recommended article: Optimal control for the coefficients of a quasilinear parabolic equation
Recommended article: DIFFERENCE APPROXIMATION AND REGULARIZATION OF THE OPTIMAL CONTROL PROBLEM FOR A PARABOLIC EQUATION WITH AN INTEGRAL CONDITION
description: scientific article; zbMATH DE number 584216